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Chapter 5: Problem 31

A man has six shirts and four ties. Assuming that they all match, how many different shirt-and-tie combinations can he wear?

Short Answer

Expert verified
There are 24 combinations.

Step by step solution

01

- Understand the Problem

We are given the number of shirts and the number of ties a man has, and we need to determine the different combinations of shirts and ties he can wear.
02

- Identify the Numbers

The man has 6 shirts and 4 ties.
03

- Use the Multiplication Principle

To find the total number of combinations, we multiply the number of shirts by the number of ties.
04

- Perform the Calculation

Calculate the number of combinations by multiplying 6 (shirts) by 4 (ties). \( 6 \times 4 = 24 \)
05

- State the Result

There are 24 different shirt-and-tie combinations that the man can wear.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

multiplication principle
The multiplication principle is a fundamental concept in combinatorics. It helps us determine the total number of outcomes when combining two or more independent choices.
For example, in our exercise, choosing a shirt is independent of choosing a tie.
We have 6 options for shirts and 4 options for ties.
  • First, we select one of the 6 shirts.
  • Next, we choose one of the 4 ties for each shirt.
To find the total number of possible combinations, multiply the number of shirts by the number of ties:
\( 6 \times 4 = 24 \)
This means there are 24 unique shirt-and-tie combinations. The multiplication principle simplifies our task by breaking it into smaller, manageable parts.
combinations
In mathematics, combinations refer to the different ways of selecting items from a group without regard to the order of selection.
In our exercise, a combination consists of one shirt and one tie.
We are interested in how many unique sets we can form given the items available.
Each choice of shirt can be paired uniquely with each choice of tie.
  • This gives us each possible pairing as a separate combination.
If we had 6 shirts and 4 ties, then each shirt can be paired with any of the 4 ties, resulting in:
\( 6 \) shirts \( \times 4 \) ties = \( 24 \) combinations.
Thus, combinations in this context are the different pairings of the shirts and ties.
basic counting principles
Basic counting principles are essential for solving problems in combinatorics.
The two main principles include the addition principle and the multiplication principle.
  • The Addition Principle: If you have two groups and need to find the total number of options, you simply add them. However, this principle doesn鈥檛 apply to our exercise as we鈥檙e multiplying options of independent choices.
  • The Multiplication Principle: This principle states that if there are multiple stages of choices, you multiply the number of choices at each stage to find the total number of combinations.
In our problem, we used the multiplication principle.
Basic counting principles provide the foundation for understanding more complex problems in combinatorics, like permutations and combinations. Mastering these basics helps simplify bigger problems into smaller, manageable parts.

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Most popular questions from this chapter

According to the Sefton Council Password Policy (August 2007), the United Kingdom government recommends the use of 鈥淓nviron passwords with the following format: consonant, vowel, consonant, consonant, vowel, consonant, number, number (for example, pinray45).鈥� (a) Assuming passwords are not case sensitive, how many such passwords are possible (assume that there are 5 vowels and 21 consonants)? (b) How many passwords are possible if they are case sensitive?

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My wife has organized a monthly neighborhood party. Five people are involved in the group: Yolanda (my wife), Lorrie, Laura, Kim, and Anne Marie. They decide to randomly select the first and second home that will host the party. What is the probability that my wife hosts the first party and Lorrie hosts the second? Note: Once a home has hosted, it cannot host again until all other homes have hosted.

A National Ambulatory Medical Care Survey administered by the Centers for Disease Control found that the probability a randomly selected patient visited the doctor for a blood pressure check is \(0.593 .\) The probability a randomly selected patient visited the doctor for urinalysis is 0.064. Can we compute the probability of randomly selecting a patient who visited the doctor for a blood pressure check or urinalysis by adding these probabilities? Why or why not?

Go to www.pearsonhighered.com/sullivanstats to obtain the data file SullivanStatsSurveyI using the file format of your choice for the version of the text you are using. The data represent the results of a survey conducted by the author. The variable "Text while Driving" represents the response to the question, "Have you ever texted while driving?" The variable "Tickets" represents the response to the question, "How many speeding tickets have you received in the past 12 months?" Treat the individuals in the survey as a random sample of all U.S. drivers. (a) Build a contingency table treating "Text while Driving" as the row variable and "Tickets" as the column variable. (b) Determine the marginal relative frequency distribution for both the row and column variable. (c) What is the probability a randomly selected U.S. driver texts while driving? (d) What is the probability a randomly selected U.S. driver received three speeding tickets in the past 12 months? (e) What is the probability a randomly selected U.S. driver texts while driving or received three speeding tickets in the past 12 months?
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